A circle of a sphere is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
that lies on a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
. Such a circle can be formed as the intersection of a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
and a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, or of two spheres. Circles of a sphere are the
spherical geometry analogs of
generalised circle
In geometry, a generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and ...
s in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. A circle on a sphere whose plane passes through the center of the sphere is called a ''
great circle'', analogous to a Euclidean
straight line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
; otherwise it is a small circle, analogous to a Euclidean circle. Circles of a sphere have radius less than or equal to the sphere radius, with equality when the circle is a great circle.
A circle of a sphere can also be characterized as the locus of points on the sphere at uniform distance from a given center point, or as a spherical
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
of constant
curvature.
On the earth
In the
geographic coordinate system on a globe, the
parallels of
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
are small circles, with the
Equator the only great circle. By contrast, all
meridians of
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
, paired with their opposite meridian in the other
hemisphere
Hemisphere refers to:
* A half of a sphere
As half of the Earth
* A hemisphere of Earth
** Northern Hemisphere
** Southern Hemisphere
** Eastern Hemisphere
** Western Hemisphere
** Land and water hemispheres
* A half of the (geocentric) celes ...
, form great circles.
Related terminology
The diameter of the sphere which passes through the center of the circle is called its axis and the endpoints of this diameter are called its poles. A circle of a sphere can also be defined as the set of points at a given
angular distance
Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle between the two sightlines, or between two point objects as viewed from an observer.
Angular distance appears in mathematics (in par ...
from a given pole.
Sphere-plane intersection
When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows:
Let ''S'' be a sphere with center ''O'', ''P'' a plane which intersects ''S''. Draw perpendicular to ''P'' and meeting ''P'' at ''E''. Let ''A'' and ''B'' be any two different points in the intersection. Then ''AOE'' and ''BOE'' are right triangles with a common side, ''OE'', and hypotenuses ''AO'' and ''BO'' equal. Therefore, the remaining sides ''AE'' and ''BE'' are equal. This proves that all points in the intersection are the same distance from the point ''E'' in the plane ''P'', in other words all points in the intersection lie on a circle ''C'' with center ''E''. This proves that the intersection of ''P'' and ''S'' is contained in ''C''. Note that ''OE'' is the axis of the circle.
Now consider a point ''D'' of the circle ''C''. Since ''C'' lies in ''P'', so does ''D''. On the other hand, the triangles ''AOE'' and ''DOE'' are right triangles with a common side, ''OE'', and legs ''EA'' and ''ED'' equal. Therefore, the hypotenuses ''AO'' and ''DO'' are equal, and equal to the radius of ''S'', so that ''D'' lies in ''S''. This proves that ''C'' is contained in the intersection of ''P'' and ''S''.
As a corollary, on a sphere there is exactly one circle that can be drawn through three given points.
The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.
[Hobbs, Prop. 310]
Compare also
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
s, which can produce
oval
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one ...
s.
Sphere-sphere intersection
To show that a non-trivial intersection of two spheres is a circle, assume (without loss of generality) that one sphere (with radius
) is centered at the origin. Points on this sphere satisfy
:
Also without loss of generality, assume that the second sphere, with radius
, is centered at a point on the positive x-axis, at distance
from the origin. Its points satisfy
:
The intersection of the spheres is the set of points satisfying both equations. Subtracting the equations gives
:
In the singular case
, the spheres are concentric. There are two possibilities: if
, the spheres coincide, and the intersection is the entire sphere; if
, the spheres are disjoint and the intersection is empty.
When ''a'' is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres.
The result follows from the previous proof for sphere-plane intersections.
See also
*
Line-plane intersection
*
Line–sphere intersection
In analytic geometry, a line and a sphere can intersect in three ways:
# No intersection at all
# Intersection in exactly one point
# Intersection in two points.
Methods for distinguishing these cases, and determining the coordinates for the ...
References
*
Further reading
*{{cite book , title=Solid Geometry
, url=https://archive.org/details/solidgeometry01comsgoog
, first1=M., last1=Sykes, first2=C.E., last2=Comstock, publisher=Rand McNally
, year=1922, page
81ff
Rotational symmetry
Spherical curves
Circles